Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if parabola and $<0$ if it is an ellipse.
But my confusion is that they do not include pair of straight lines and the circle(though it is a special case of the ellipse).
Is this what you want?
Let $p=B^2-4AC$.
If $p\lt 0$, ellipse, circle, point or no curve.
If $p=0$, parabola, 2 parallel lines, 1 line or no curve.
If $p\gt 0$, hyperbola or 2 intersecting lines.
more information here with figures.
$1. $ if it is a circle $$A=C$$ and $B=0$
$2.$ if it is a pair of straight line $$B^2-4AC\ge 0$$ and $$ACK+\frac{BDE}{4}-\frac{AE^2}{4}-\frac{CD^2}{4}-\frac{KB^2}{4}=0$$
